THE GREAT ALGORIST
that Jacob! made Ms great start in entire ignorance of bis
rival's work.
A capital property of the elliptic functions is their double
periodicity (discovered in 1825 by Abel): if E(x) is an elliptic
function, then there are two distinct numbers, say pv p2, such
that
E(x + pi) = E(x), and Efa
for all values of the variable #,
Finally, on the historical side, is the somewhat tragic part
played by Legendre. For forty years he had slaved over elliptic
integrals (not elliptic functions) without noticing what both
Abel and Jacobi saw almost at once, namely that by inverting
his point of view the whole subject would become mfinitely
simpler. Elliptic integrals first present themselves in the pro-
blem of finding the length of an arc of an ellipse. To what was
said about inversion in connexion with Abel the following
statement in symbols may be added. This will bring out more
clearly the point which Legendre missed,
If R(t ) denotes a polynomial in t , an integral of the type
1
r
J (
is called an elliptic integral if E(t) is of either the third or the
fourth degree; if R(t) is of degree higher than the fourths the
integral is called Abelian (after Abel, some of whose greatest
work concerned such integrals). If R(t) is of only the second
degree, the integral can be calculated out in terms of elementary
functions. In particular
f* 1
— dt
J o Vl -
(sin-% is read, 'an angle whose sine is a?*). That is, if
we consider the upper limit, a, of the integral, as a function of
the integral itself, namely of y. This inversion of the problem
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